Accurate Computations Of Lupas-like Totally Nonpositive Matrices

Some achievements have been made in Demmel and Kahan high-precision calculation of double-diagonal matrix,which has led many people to dig into the problem of high-precision calculation of structural matrix.Particularly Koev double-diagonal decomposition of non-negative matrices using Neville elimination method is used to find out the inverse,linear system,LDU decomposition and high-precision calculation of eigenvalues and singular values.Compared to completely non-positive high-precision calculation of singular values,linear systems,LDU decomposition and high-precision calculation of eigenvalues are in the bud,and now completely non-positive matrices have many important applications in big data processing,precision instrument design and computer-aided geometry design.Therefore,it is of great practical significance to explore completely non-positive structural matrices.This paper first draws out Lupas class matrix by interpolation problem,then uses the Neville elimination method to decompose the research matrix by double diagonal,realizes the parameterization of the matrix(because these parameters are effectively improved by the quotient of some sub-forms of the matrix because the numerical operations cancel each other and guarantee the accuracy of the results),and verifies that the matrix and the inverse matrix are completely non-positive matrices,and then designs the corresponding algorithm by using the parameters obtained by the bidiagonal decomposition to solve the matrix eigenvalues,singular values and solve the linear system.The results obtained are compared with the singular values obtained by Mathmatic and the singular values obtained by Matlab,and the results obtained by the algorithm have better accuracy.? In the first chapter,guided by the theoretical background of the complete non-negative matrix,derives that the study of the complete non-positive matrix hasimportant applications in big data processing,precision instrument design andcomputer-aided geometric design.The correlation theory and properties of thecomplete non-negative matrix are explored by the complete non-negative matrixcorrelation theory.Then the research status of the complete non-positive matrixand the sub-form of the Lupas matrix are given.? In the second chapter,defines Lupas class matrix by Lupas matrix linear systemproblem,and then verifies that the parameters of the matrix satisfy the necessaryand sufficient conditions of the judgment theorem of the complete non-positivematrix Lupas prove that the class matrix is a complete non-positive matrix.Atthe same time,we give the theorem that the inverse Lupas class matrix hasunique decomposition and is completely non-positive matrix.Finally,a high-precision algorithm and program are designed to solve the singular value ofLupas matrix and the related problems of linear system.? In the third chapter,defines the generalized Lupas matrix and calculates therelation between the generalized Lupas matrix sub-form and the Lupas matrixsub-form.Finally,a high-precision algorithm is designed to solve the singu-lar value and eigenvalue problems of the matrix.Finally,the high-precisionalgorithm is verified by numerical experiments.